Inference for ordinary least squares, lasso/NG, horseshoe and ridge regression models by (Gibbs) sampling from the Bayesian posterior distribution, augmented with Reversible Jump for model selection
bhs(X, y, T=1000, thin=NULL, RJ=TRUE, M=NULL, beta=NULL,
lambda2=1, s2=var(y-mean(y)), mprior=0, ab=NULL,
theta=0, rao.s2=TRUE, icept=TRUE, normalize=TRUE, verb=1)
bridge(X, y, T = 1000, thin = NULL, RJ = TRUE, M = NULL,
beta = NULL, lambda2 = 1, s2 = var(y-mean(y)), mprior = 0,
rd = NULL, ab = NULL, theta=0, rao.s2 = TRUE, icept = TRUE,
normalize = TRUE, verb = 1)
blasso(X, y, T = 1000, thin = NULL, RJ = TRUE, M = NULL,
beta = NULL, lambda2 = 1, s2 = var(y-mean(y)),
case = c("default", "ridge", "hs", "ng"), mprior = 0, rd = NULL,
ab = NULL, theta = 0, rao.s2 = TRUE, icept = TRUE,
normalize = TRUE, verb = 1)blasso returns an object of class "blasso", which is a
list containing a copy of all of the input arguments as well as
of the components listed below.
a copy of the function call as used
a vector of T samples of the (un-penalized)
“intercept” parameter
a T*ncol(X) matrix of T samples from
the (penalized) regression coefficients
the number of non-zero entries in each vector of T
samples of beta
a vector of T samples of the variance parameter
a vector of T samples of the penalty
parameter
a vector of T with the gamma parameter
when case = "ng"
a T*ncol(X) matrix of T samples from
the (latent) inverse diagonal of the prior covariance matrix for
beta, obtained for Lasso regressions
a T*nrow(X) matrix of T samples
from the (latent) diagonal of the covariance matrix of the response
providing a scale-mixture implementation of Student-t errors with
degrees of freedom nu when active (input theta > 0)
a vector of T samples of the degrees of freedom
parameter to the Student-t errors mode when active
(input theta > 0)
a vector of T samples of the Binomial proportion
p that was given a Beta prior, as described above for the
2-vector version of the mprior input
the log posterior probability of each (saved) sample of the joint parameters
the log likelihood of each (saved) sample of the parameters
the log likelihood of each (saved) sample of the
parameters under the Normal errors model when sampling under the
Student-t model; i.e., it is not present
unless theta > 0
data.frame, matrix, or vector of inputs X
vector of output responses y of length equal to the
leading dimension (rows) of X, i.e., length(y) == nrow(X)
total number of MCMC samples to be collected
number of MCMC samples to skip before a sample is
collected (via thinning). If NULL (default), then
thin is determined based on the regression model implied
by RJ, lambda2, and ncol(X); and also
on the errors model implied by theta and nrow(X)
if TRUE then model selection on the columns of the
design matrix (and thus the parameter beta in the model) is
performed by Reversible Jump (RJ) MCMC. The initial model is
specified by the beta input, described below, and the maximal
number of covariates in the model is specified by M
the maximal number of allowed covariates (columns of
X) in the model. If input lambda2 > 0 then any
M <= ncol(X) is allowed. Otherwise it must be that
M <= min(ncol(X), length(y)-1), which is default value
when a NULL argument is given
initial setting of the regression coefficients. Any
zero-components will imply that the corresponding covariate (column
of X) is not in the initial model. When input RJ =
FALSE (no RJ) and lambda2 > 0 (use lasso) then no
components are allowed to be exactly zero. The default setting is
therefore contextual; see below for details
square of the initial lasso penalty parameter. If zero, then least squares regressions are used
initial variance parameter
specifies if ridge regression, the
Normal-Gamma, or the horseshoe prior should be done instead
of the lasso; only meaningful when lambda2 > 0
prior on the number of non-zero regression coefficients
(and therefore covariates) m in the model. The default
(mprior = 0) encodes the uniform prior on 0 <= m <= M.
A scalar value 0 < mprior < 1 implies a Binomial prior
Bin(m|n=M,p=mprior). A 2-vector mprior=c(g,h)
of positive values g and h represents
gives Bin(m|n=M,p) prior where p~Beta(g,h)
=c(r, delta), the alpha (shape) parameter and
\(\beta\) (rate) parameter to the gamma distribution prior
G(r,delta) for the \(\lambda^2\) parameter under
the lasso model; or, the \(\alpha\) (shape) parameter and
\(\beta\) (scale) parameter to the
inverse-gamma distribution IG(r/2, delta/2) prior for
the \(\lambda^2\)
parameter under the ridge regression model. A default of NULL
generates appropriate non-informative values depending on the
nature of the regression. Specifying rd=FALSE causes
lambda2 values to be fixed at their starting value, i.e., not
sampled. See the details below for information
on the special settings for ridge regression
=c(a, b), the \(\alpha\) (shape)
parameter and the \(\beta\) (scale) parameter for the
inverse-gamma distribution prior IG(a,b) for the variance
parameter s2. A default of NULL generates appropriate
non-informative values depending on the nature of the regression
the rate parameter (> 0) to the exponential prior
on the degrees of freedom paramter nu under a model with
Student-t errors implemented by a scale-mixture prior.
The default setting of theta = 0 turns off this prior,
defaulting to a normal errors prior
indicates whether Rao-Blackwellized samples for
\(\sigma^2\) should be used (default TRUE); see
below for more details
if TRUE, an implicit intercept term is fit
in the model, otherwise the the intercept is zero; default is
TRUE
if TRUE, each variable is standardized
to have unit L2-norm, otherwise it is left alone; default is
TRUE
verbosity level; currently only verb = 0 and
verb = 1 are supported
Robert B. Gramacy rbg@vt.edu
The Bayesian lasso model and Gibbs Sampling algorithm is described
in detail in Park & Casella (2008). The algorithm implemented
by this function is identical to that described therein, with
the exception of an added “option” to use a Rao-Blackwellized
sample of \(\sigma^2\) (with \(\beta\) integrated out)
for improved mixing, and the model selections by RJ described below.
When input argument lambda2 = 0 is
supplied, the model is a simple hierarchical linear model where
\((\beta,\sigma^2)\) is given a Jeffrey's prior
Specifying RJ = TRUE causes Bayesian model selection and
averaging to commence for choosing which of the columns of the
design matrix X (and thus parameters beta) should be
included in the model. The zero-components of the beta input
specify which columns are in the initial model, and
M specifies the maximal number of columns.
The RJ mechanism implemented here for the Bayesian lasso model selection differs from the one described by Hans (2009), which is based on an idea from Geweke (1996). Those methods require departing from the Park & Casella (2008) latent-variable model and requires sampling from each conditional \(\beta_i | \beta_{(-i)}, \dots\) for all \(i\), since a mixture prior with a point-mass at zero is placed on each \(\beta_i\). Out implementation here requires no such special prior and retains the joint sampling from the full \(\beta\) vector of non-zero entries, which we believe yields better mixing in the Markov chain. RJ proposals to increase/decrease the number of non-zero entries does proceed component-wise, but the acceptance rates are high due due to marginalized between-model moves (Troughton & Godsill, 1997).
When the lasso prior or RJ is used, the automatic thinning level
(unless thin != NULL) is determined by the number of columns
of X since this many latent variables are introduced
Bayesian ridge regression is implemented as a special case via the
bridge function. This essentially calls blasso
with case = "ridge". A default setting of rd = c(0,0) is
implied by rd = NULL, giving the Jeffery's prior for the
penalty parameter \(\lambda^2\) unless ncol(X) >=
length(y) in which case the proper specification of rd =
c(5,10) is used instead.
The Normal--Gamma prior (Griffin & Brown, 2009) is implemented as
an extension to the Bayesian lasso with case = "ng". Many
thanks to James Scott for providing the code needed to extend the
method(s) to use the horseshoe prior (Carvalho, Polson, Scott, 2010).
When theta > 0 then the Student-t errors via scale mixtures
(and thereby extra latent variables omega2) of Geweke (1993)
is applied as an extension to the Bayesian lasso/ridge model.
If Student-t errors are used the automatic thinning level
is augmented (unless thin != NULL) by the number of rows
in X since this many latent variables are introduced
Park, T., Casella, G. (2008). The Bayesian Lasso.
Journal of the American Statistical Association, 103(482),
June 2008, pp. 681-686
tools:::Rd_expr_doi("10.1198/016214508000000337")
Griffin, J.E. and Brown, P.J. (2009).
Inference with Normal-Gamma prior distributions in
regression problems. Bayesian Analysis, 5, pp. 171-188.
tools:::Rd_expr_doi("10.1214/10-BA507")
Hans, C. (2009). Bayesian Lasso regression.
Biometrika 96, pp. 835-845.
tools:::Rd_expr_doi("10.1093/biomet/asp047")
Carvalho, C.M., Polson, N.G., and Scott, J.G. (2010) The
horseshoe estimator for sparse signals. Biometrika 97(2):
pp. 465-480.
https://faculty.chicagobooth.edu/nicholas.polson/research/papers/Horse.pdf
Geweke, J. (1996). Variable selection and model comparison in regression. In Bayesian Statistics 5. Editors: J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, 609-620. Oxford Press.
Paul T. Troughton and Simon J. Godsill (1997). A reversible jump sampler for autoregressive time series, employing full conditionals to achieve efficient model space moves. Technical Report CUED/F-INFENG/TR.304, Cambridge University Engineering Department.
Geweke, J. (1993) Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics, Vol. 8, S19-S40
## following the lars diabetes example
data(diabetes)
attach(diabetes)
## Ordinary Least Squares regression
reg.ols <- regress(x, y)
## Lasso regression
reg.las <- regress(x, y, method="lasso")
## Bayesian Lasso regression
reg.blas <- blasso(x, y)
## summarize the beta (regression coefficients) estimates
plot(reg.blas, burnin=200)
points(drop(reg.las$b), col=2, pch=20)
points(drop(reg.ols$b), col=3, pch=18)
legend("topleft", c("blasso-map", "lasso", "lsr"),
col=c(2,2,3), pch=c(21,20,18))
## plot the size of different models visited
plot(reg.blas, burnin=200, which="m")
## get the summary
s <- summary(reg.blas, burnin=200)
## calculate the probability that each beta coef != zero
s$bn0
## summarize s2
plot(reg.blas, burnin=200, which="s2")
s$s2
## summarize lambda2
plot(reg.blas, burnin=200, which="lambda2")
s$lambda2
if (FALSE) {
## fit with Student-t errors
## (~400-times slower due to automatic thinning level)
regt.blas <- blasso(x, y, theta=0.1)
## plotting some information about nu, and quantiles
plot(regt.blas, "nu", burnin=200)
quantile(regt.blas$nu[-(1:200)], c(0.05, 0.95))
## Bayes Factor shows strong evidence for Student-t model
mean(exp(regt.blas$llik[-(1:200)] - regt.blas$llik.norm[-(1:200)]))
}
## clean up
detach(diabetes)
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